Intro and Simulation

Why are we Here?

Let’s do some time traveling

Modern Tools

Maybe …

The best we have today (I think)

Get a Julius account

• Free account has very strict message limit
• Basic allows 250 messages per month. Maybe enough?
• Essential allows unlimited messages.
• 50% academic discount. Sign up using your Rice email and enter the promo code STUDY2024.

What Julius does

• Writes, executes, and debugs python
• Provides links to images, tables, datasets it creates
• Provides download of code (and conversation) as Jupyter notebook
• Allows us to upload our own data
• Can get some online data
• Can install any python libraries not pre-installed

Warm-Up

• Ask Julius to plot the function \(y=x^2\)
• Ask Julius to plot the payoff diagram of a call option with a strike of 100.
• Ask Julius to get the history of 10-year Treasury yields from FRED and plot them.
• Ask Julius to simulate 30 returns from a normal distribution with mean of 10% and standard deviation of 15% and plot the cumulative compound return.

Google Colab

• Download the chat as a notebook.
  
• Go to https://colab.research.google.com/ and sign in.
• Use File/Open Notebook to open the saved notebook.
• Run the notebook cell by cell or use Runtime/Run All.

• If you want to save figures or tables created by a notebook in Colab, you can
  
• mount your Google drive (folder icon on left and then Google drive icon in the top toolbar on left)
• then execute code to save to “drive/MyDrive/filename.ext”
• To install additional libraries, execute, e.g., !pip install yfinance in a code cell.

Today’s Chats

• Long-run risk
• Retirement planning
• Retirement planning with simulation
• Monte-Carlo option valuation
• Monte-Carlo enterprise valuation

Long-Run Risk

• Ask Julius to simulate how much a $1 investment would grow to in 10 years if the investment return is normally distributed with a mean of 6% and a standard deviation of 20%.
• Ask Julius to calculate the fraction of times the investment outperforms a risk-free return of 1% over 10 years.
• Ask Julius to generate boxplots and histograms of the ending investment account balance.

Retirement Planning

• Tell Julius you want to check if a retirement savings plan is feasible.
  
• Ask Julius what information you need to provide and provide it.
• Ask Julius to calculate the ending balance as a function of the rate of return over some range and plot it.

Retirement Planning with Simulation

• Ask Julius to simulate the retirement plan assuming the annual returns are normally distributed with some mean and variance.
• Ask Julius to describe the distribution of ending account balances and to produce a boxplot and histogram.

Monte-Carlo Option Valuation

• Tell Julius you want to value a European call option by Monte Carlo. Ask Julius what information you need to provide and provide it.
• Ask Julius to value the same call option using Black-Scholes.
• Ask Julius to value a put option both ways.

Enterprise Valuation

• Give Julius the following data (from Applied):
  • A firm with no debt will have free cash flow of 100M next year.
  • The cash flow will grow by 12% per year for years 2 through 5.
  • Afterwards the cash flow will grow by 3% per year forever.
  • The firm has a 10% cost of capital and there are 44.75M shares outstanding.
• Ask Julius what the share price should be.

Check for Errors

• It is quite possible that Julius will make timing errors.
• You can edit the code to fix any errors.
• Check the following:
  • There should be five cash flows in the first stage.
  • The first cash flow should be 100M (no growth).
  • The terminal value should be 100M with 4 years of growth at 12% and 1 year of growth at 3% divided by (10% - 3%).
  • The terminal value should be discounted back 5 years.

Sensitivity Analysis

• Ask Julius to vary the first stage growth rate between 6% and 18% and to plot the share price as a function of the growth rate.

Monte-Carlo Enterprise Valuation I

• Now ask Julius to simulate the first-stage growth rate from a normal distribution with a mean of 12% and a standard deviation of 3%.
  
• Ask Julius to compute the share price in each simulation and to describe the share price distribution.
• Ask Julius to produce a histogram of the share price distribution.

Monte-Carlo Enterprise Valuation II

• TBD